Integrable anisotropic spin-ladder model

PHYSICAL REVIEW B, VOLUME 64, 054420

Integrable anisotropic spin-ladder model

Arlei Prestes Tonel,1,* Angela Foerster,1,† Jon Links,1,2,‡ and Andre Luiz Malvezzi3,§

1Instituto de Fı´sica da UFRGS, Avenida Bento Gonc¸alves 9500, Porto Alegre, RS-Brazil2Department of Mathematics, The University of Queensland, Queensland, 4072, Australia

3Departamento de Fı´sica da UNESP, Avenida Eng. Luiz Edmundo Carrijo Coube, s/n, Bauru, SP-Brazil~Received 13 December 2000; published 16 July 2001!

We present an integrable spin-ladder model, which possesses a free parameter besides the rung couplingJ.Wang’s system based on theSU(4) symmetry can be obtained as a special case. The model is exactly solvableby means of the Bethe ansatz method. We determine the dependence on the anisotropy parameter of the phasetransition between gapped and gapless spin excitations and present the phase diagram. Finally, we show thatthe model is a special case of a more general Hamiltonian with three free parameters.

DOI: 10.1103/PhysRevB.64.054420 PACS number~s!: 75.10.Jm, 71.10.Fd, 03.65.Fd

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With the discovery of high-temperature superconductivin doped copper oxide~or cuprate! materials,1 a tremendouseffort has been made to understand the physics underlthis phenomenon. In the absence of doping these compoare reasonably approximated by the two-dimensioHeisenberg model2,3 or some suitable generalization descring spin-exchange-type interaction. It is well known thatone dimension the Heisenberg model is exactly solvableBethe ansatz methods and from this solution the spectrumelementary spin excitation is gapless. On the other handexistence of the spin gap is critical for the observed phenenon of superconductivity to occur under doping. To mamize the interaction between theory and experiment, mwork is now focused on quasi-one-dimensional modknown as ladders. The introduction of these ladder systhas brought about a significantly increased understandinthe physics of the cuprate compounds.

By introducing the concept of the ladder model the appent contradiction in the excitation spectrum is resolved, sithe ladder allows for the formation of singlet states alongrungs which are responsible for the formation of the sgap. However, the usual Heisenberg ladder model cannosolved. In order to gain some results in the theory of sladder systems, many authors have considered generamodels, which incorporate additional interaction terms tguarantee exact solvability. Remarkably, such generalmodels still exhibit realistic physical properties such asexistence of a spin gap4 and the magnetization plateausfractional values of the total magnetization.5

This approach has been used to derive quasi-odimensional systems using the well-established theories fthe one-dimensional case.4–15 In all cases cited above, nfree parameters are present, other than the rung interacoupling and applied magnetic field, due to the strict contions of integrability. With the presence of free parameteris reasonable to expect that the solution may provide betest models for describing the various behaviors associwith ladder systems.

The purpose of this paper is to present an integrable geralized spin ladder with one extra parameter, characterianisotropy, without violating integrability. This model is eactly solvable by the Bethe ansatz and it reduces tomodel introduced by Wang4 for a special limit of this extraparameter. The situation here is akin to the generalizatio

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theXXX chain to the anisotropicXXZ version. The introduc-tion of the additional free parameter in the present caselows for an example of a model with a critical line varyincontinuously with the anisotropy. More specifically, the siof the gap in the massive region depends explicitly onanisotropy parameter, which in turn shows dependence oanisotropy parameter for the points at which the gap clothat define the phase transition.

Let us begin by introducing the generalized spin-laddmodel, whose Hamiltonian reads

H (1)5(j 51

L Fhj , j 1111

2J~sW j•tW j21!G , ~1!

where

hj , j 115 14 ~11s j

zs j 11z !~11t j

zt j 11z !1~s j

1s j 112 1s j

2s j 111 !

3~t j1t j 11

2 1t j2t j 11

1 !1 12 ~11s j

zs j 11z !~ t21t j

1t j 112

1tt j2t j 11

1 !1 12 ~ t21s j

1s j 112 1ts j

2s j 111 !

3~11t jzt j 11

z !.

AbovesW j andtW j are Pauli matrices acting on sitej of theupper and lower legs, respectively,J is the strength of therung coupling~we will consider only the caseJ.0 in thesubsequent analysis corresponding toantiferromagneticcou-pling!, andt is a free parameter representing an anisotropythe legs and interchain interaction. Throughout,L is the num-ber of rungs~equivalently, the length of the ladder! and pe-riodic boundary conditions are imposed. By settingt→1 inEq. ~1!, Wang’s model based on theSU(4) symmetry4 canbe recovered.~Strictly speaking, it isSU(4) invariant in theabsence of the rung interactions.! The Hamiltonian is invari-ant under interchange of the legs; i.e.,sW j↔tW j . Moreover,under spin inversion for both leg spaces the Hamiltonianinvariant with the interchanget↔t21. For this reason we sethat the parametert plays the role of spin anisotropy.

The energy eigenvalues of the Hamiltonian are given

E52(j 51

M1 S 1

l j211/4

22JD 1~122J!L, ~2!

©2001 The American Physical Society20-1

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TONEL, FOERSTER, LINKS, AND MALVEZZI PHYSICAL REVIEW B64 054420

FIG. 1. EnergiesEi ( i 50, . . . ,8)versus rung couplingJ for different values of the anisotropyt. Notice that there is mainly competitionbetweenE0 andE1 to be the lowest-energy level of the model. In addition, the critical value ofJ above whichE0 is the ground-state energvaries witht reaching its minimum value whent51.

nth

an

rgyby

g. 1

y

incase

n

wherel j are solutions to the Bethe ansatz Eqs.~3! below.The Bethe ansatz equations arise from the exact solutiothe model through the nested algebraic Bethe ansatz meand read

t (L22M3)S l j2 i /2

l j1 i /2D L

5)lÞ j

M1 l j2l l2 i

l j2l l1 i )a51

M2 l j2ma1 i /2

l j2ma2 i /2,

t (L22M3) )bÞa

M2 ma2mb2 i

ma2mb1 i5)

j 51

M1 ma2l j2 i /2

ma2l j1 i /2

3 )d51

M3 ma2nd2 i /2

ma2nd1 i /2, ~3!

t (L12M222M1) )gÞd

M3 nd2ng2 i

nd2ng1 i5 )

a51

M2 nd2ma2 i /2

nd2ma1 i /2.

We remark that although the anisotropy parametert doesnot appear explicitly in the energy expression~2!, the solu-tionsl j for the Bethe ansatz equation do depend ont as willbe illustrated later.

The exact diagonalization of the two-site Hamiltonishows that forJ.11 1

2 (t11/t) the ~unique! ground state

05442

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assumes the form of the product of the singlets with eneE05224J and the energies of the excitations are given

E1522J2~ t11/t !, E2522J1~ t11/t !,

E35~ t11/t !, E452~ t11/t !,

E5522J12, E6522J22, E752, E8522.

A sample of these numerical results are presented in Fiabove.

For L sites it follows that the ground state is still given ba product of rung singlets whenJ.11 1

2 (t11/t) and theenergy is (122J)L. This is in fact the reference state usedthe Bethe ansatz calculation and corresponds to theM15M25M350 for the Bethe ansatz Eq.~3!. To describean elementary spin-1 excitation, we takeM151 and M25M350 in Eq. ~3!, which leads to the imaginary solutiofor the variablel ~strictly, the lattice lengthL is assumed tobe even!,

l5i

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t11, ~4!

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INTEGRABLE ANISOTROPIC SPIN-LADDER MODEL PHYSICAL REVIEW B64 054420

giving the minimal excited-state energy. The energy gapeasily be calculated using the exact Bethe ansatz solutionhas the form

D52S J2121

2 S t11

t D D . ~5!

By solving D50 for J we find the critical valueJc511 1

2 (t11/t), indicating the critical line at which the quantumphase transition from the dimerized phase to the gapphase occurs.

The phase diagram in Fig. 2~a! assumes a simpler formafter a suitable reparametrization. We introduce a newrameterK given byK5(t11/t)/2>1. In Fig. 2~b! the phasediagram is represented in terms ofK andJ. The phase boundary is now a straight line given byJ511K. As a prelimi-nary attempt to characterize the gapless phase we have

FIG. 2. ~a! Rung couplingJ versus anisotropyt. This graphicrepresents the phase diagram and the dotted line shows Wpoint. The curve@J511(t11/t)/2# divides the gapped and gaplephases.~b! Rung couplingJ versus reparametrization parameterK.This graphic shows a reparametrization of the curveJ511(t11/t)/2 in terms of K5(t11/t)/2. In this parameterization, thphase boundary is a straight line.

05442

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ied by numerical diagonalization the energy spectra of~1! on ladders of sizes up to eight rungs and several valueJ andK.

As mentioned above, ifJ.11K the spectra is gappeand the ground state a product of singlets on each rungwe cross the phase transition line a state with finite magtization becomes the ground state. By further decreasinJ,with K fixed, the ground-state magnetization initially increases and then, ifJ is made small enough, drops to zerThis behavior resembles that of the one-dimensional antropic Heisenberg model, also called theXXZ chain, in thepresence of a magnetic field.16 In the XXZ chain with nomagnetic field applied, the anisotropy can be tuned to brthe system into a massive antiferromagnetic~AF! phase,where the ground state is a Ne´el state. By tuning the magnetic field inside this AF phase we observe a behavior simiar to the one found here in terms of the parameterJ. Inparticular, the Pokrovsky-Talapov16,17 phase transition appearing in the phase diagram of theXXZ chain has manyfeatures in common with the phase transition found hewith J andK playing the roles of the magnetic field and thanisotropy, respectively. Based on this analogy, we conture that the ground state with null magnetization foundour model for small values ofJ suggests the presence of aAF gapped phase~Neel phase!. Therefore, another phasetransition line is expected to exist below the one presenhere.

The integrability of this model can be shown by the fathat it can be mapped@see Eq.~7! below# to the followingHamiltonian, which can be derived from anR matrix obey-ing the Yang-Baxter algebra forJ50, while for JÞ0 therung interactions take the form of a chemical-potential ter

H (1)5(j 51

L

@ h j , j 1122JXj00#, ~6!

where

h j , j 115 (a50

3

XjaaXj 11

aa 1Xj20Xj 11

02 1Xj02Xj 11

20 1Xj13Xj 11

31

1Xj31Xj 11

13 1t~Xj10Xj 11

01 1Xj12Xj 11

21 1Xj03Xj 11

30

1Xj23Xj 11

32 !1t21~Xj01Xj 11

10 1Xj21Xj 11

12 1Xj30Xj 11

03

1Xj32Xj 11

23 !.

Above Xjab5ua j&^b j u are the Hubbard operators wit

ua j& the orthogonalized eigenstates of the local operatorsW j

•tW j , as in Wang’s case.4 The local Hamiltonians~1! and ~6!are related through the following basis transformation:

u↑,↑&→1/A2~ u↑,↓&2u↓,↑&), u↑,↓&→u↑,↑&,~7!

u↓,↑&→1/A2~ u↑,↓&1u↓,↑&), u↓,↓&→u↓,↓&.

The following R matrix,

g’s

0-3

TONEL, FOERSTER, LINKS, AND MALVEZZI PHYSICAL REVIEW B64 054420

R5

¨

a 0 0 0 u 0 0 0 0 u 0 0 0 0 u 0 0 0 0

0 t21b 0 0 u c 0 0 0 u 0 0 0 0 u 0 0 0 0

0 0 b 0 u 0 0 0 0 u c 0 0 0 u 0 0 0 0

0 0 0 tb u 0 0 0 0 u 0 0 0 0 u c 0 0 0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 c 0 0 u tb 0 0 0 u 0 0 0 0 u 0 0 0 0

0 0 0 0 u 0 a 0 0 u 0 0 0 0 u 0 0 0 0

0 0 0 0 u 0 0 tb 0 u 0 c 0 0 u 0 0 0 0

0 0 0 0 u 0 0 0 b u 0 0 0 0 u 0 c 0 0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 c 0 u 0 0 0 0 u b 0 0 0 u 0 0 0 0

0 0 0 0 u 0 0 c 0 u 0 t21b 0 0 u 0 0 0 0

0 0 0 0 u 0 0 0 0 u 0 0 a 0 u 0 0 0 0

0 0 0 0 u 0 0 0 0 u 0 0 0 tb u 0 0 c 0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 c u 0 0 0 0 u 0 0 0 0 u t21b 0 0 0

0 0 0 0 u 0 0 0 c u 0 0 0 0 u 0 b 0 0

0 0 0 0 u 0 0 0 0 u 0 0 0 c u 0 0 t21b 0

0 0 0 0 u 0 0 0 0 u 0 0 0 0 u 0 0 0 a

©, ~8!

aseters

with

a5x11, b5x, and c51

obeys the Yang-Baxter algebra

R12~x2y!R13~x!R23~y!5R23~y!R13~x!R12~x2y! ~9!

and originates the Hamiltonian~6! for J50 by the standardprocedure18

h j , j 115Pd

dxR~x!ux50 ,

05442

whereP is the permutation operator.This model studied above represents one particular c

of a more general Hamiltonian that has three free parameand reads

Hg5(j 51

L Fhj , j 11g 1

1

2J~sW j•tW j21!G , ~10!

where

hj , j 11g 5s j

1s j 112 F t1

21

4~11t j

z!~11t j 11z !1

t2

4~12t j

z!~12t j 11z !1t3t j

1t j 112 1t j

2t j 111 G1s j

2s j 111

3F t1

4~11t j

z!~11t j 11z !1

t221

4~12t j

z!~12t j 11z !1t j

1t j 112 1t3

21t j2t j 11

1 G11

4~11s j

z!~11s j 11z !

3@ 12 ~11t j

zt j 11z !1t1

21t j1t j 11

2 1t1t j2t j 11

1 #1 14 ~12s j

z!~12s j 11z !@ 1

2 ~11t jzt j 11

z !1t2t j1t j 11

2 1t221t j

2t j 111 #.

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INTEGRABLE ANISOTROPIC SPIN-LADDER MODEL PHYSICAL REVIEW B64 054420

This Hamiltonian can be mapped to@see Eq.~7!#

Hg5(j 51

L

@ h j , j 11g 22JXj

00#, ~11!

where

h j , j 11g 5 (

a50

3

XjaaXj 11

aa 1Xj20Xj 11

02 1Xj02Xj 11

20 1t1~Xj10Xj 11

01

1Xj12Xj 11

21 !1t2~Xj30Xj 11

03 1Xj32Xj 11

23 !1t3Xj31Xj 11

13

1t121~Xj

01Xj 1110 1Xj

21Xj 1112 !1t2

21~Xj03Xj 11

30

1Xj23Xj 11

32 !1t321Xj

13Xj 1131 .

Using the algebraic nested Bethe ansatz method,model can be exactly solved. The energy eigenvalues ofHamiltonian~10! are also given by Eq.~2! while the Betheansatz equations reads

t1(L2M3)t2

M3t32M3S l j2 i /2

l j1 i /2D L

5)lÞ j

M1 l j2l l2 i

l j2l l1 i )a51

M2 l j2ma1 i /2

l j2ma2 i /2,

t1(L2M3)t2

M3t32M3 )

bÞa

M2 ma2mb2 i

ma2mb1 i

5)j 51

M1 ma2l j2 i /2

ma2l j1 i /2)d51

M3 ma2nd2 i /2

ma2nd1 i /2,

A

05442

ishe

t1(M22M1)t2

2(L2M11M2)t32(M12M2) )

gÞd

M3 nd2ng2 i

nd2ng1 i

5 )a51

M2 nd2ma2 i /2

nd2ma1 i /2.

The physics of the integrable model presented here ispected to be much richer, since the presence of these eparameters will certainly influence the phase diagram ofmodel. More details of this model are being studied andresults will be shown in a future work.

To summarize, we have introduced a generalizationWang’s spin-ladder model based on theSU(4) symmetry.This was achieved by introducing one extra parameterthe system without violating integrability. The Bethe ansequations as well as the energy expression of the modepresented. We show also that the model has a gap thapends on the free parameter and the critical point, andphase diagram was obtained. We note that our model wone free parameter is a special case of a more generalgrable Hamiltonian that has three free parameters. A comhensive analysis of the general model will be undertakenfuture work.

J.L. thanks the Fundaçao de Amparo a Pesquisa do Etado do Rio Grande do Sul and the Australian ReseaCouncil for financial support. He also thanks the InstitutoFısica da UFRGS for their kind hospitality. A.P.T., A.F. anA.L.M. thank CNPq-Conselho Nacional de Desenvolmento Cientı´fico e Tecnolo´gico for financial support. A.L.M.acknowledges the financial support from Fundaçao de Am-paro aPesquisa do Estado de Saõ Paulo~FAPESP-Brazil!.A.F. wishes to acknowledge the kind hospitality of the Ceter for Mathematical Physics at the UniversityQueensland.

c-

*Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]§Email address: [email protected]. B. Bednorz and K. A. Mu¨ller, Z. Phys. B: Condens. Matter64,

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Integrable anisotropic spin-ladder model

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